Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/212189)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by AProVE @ termCOMP 2023)

1 Dependency Pair Transformation

There are 125 ruless (increase limit for explicit display).

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

• The 1^st component contains the pair

  0#(1(2(x1)))	→	1#(x1)	(54)
  1#(0(1(2(x1))))	→	1#(x1)	(137)
  1#(0(1(2(x1))))	→	0#(x1)	(140)
  0#(1(2(x1)))	→	0#(x1)	(56)
  0#(1(2(x1)))	→	1#(0(x1))	(71)
  0#(1(2(x1)))	→	5#(x1)	(75)
  5#(0(1(2(x1))))	→	5#(x1)	(148)
  5#(0(1(2(x1))))	→	1#(5(x1))	(150)
  0#(1(2(x1)))	→	1#(5(x1))	(88)
  0#(0(1(2(x1))))	→	1#(x1)	(99)
  0#(0(1(2(x1))))	→	5#(0(x1))	(106)
  0#(0(1(2(x1))))	→	0#(x1)	(107)
  0#(1(2(0(x1))))	→	0#(0(x1))	(115)
  0#(1(2(5(2(x1)))))	→	0#(x1)	(163)
  0#(1(4(2(2(x1)))))	→	1#(1(x1))	(171)
  0#(1(4(2(2(x1)))))	→	1#(x1)	(172)

  1.1.1 Reduction Pair Processor

  Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

  [0#(x1)]	=	0 -∞ -∞ + 0 0 -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1
  [1(x1)]	=	-∞ -∞ -∞ + 0 -∞ 0 -∞ 0 1 0 -∞ -∞ · x1
  [2(x1)]	=	-∞ 0 -∞ + 0 -∞ -∞ 0 0 -∞ 1 0 1 · x1
  [1#(x1)]	=	0 -∞ -∞ + 0 -∞ 0 -∞ -∞ -∞ -∞ -∞ -∞ · x1
  [0(x1)]	=	-∞ -∞ -∞ + 0 -∞ -∞ -∞ 0 -∞ 0 -∞ -∞ · x1
  [5#(x1)]	=	0 -∞ -∞ + 0 -∞ 0 -∞ -∞ -∞ -∞ -∞ -∞ · x1
  [5(x1)]	=	-∞ -∞ -∞ + 0 -∞ 0 -∞ 0 0 0 -∞ -∞ · x1
  [4(x1)]	=	-∞ -∞ -∞ + 0 -∞ 0 0 -∞ 0 0 -∞ -∞ · x1
  [3(x1)]	=	-∞ -∞ -∞ + 0 -∞ -∞ 0 -∞ 0 -∞ -∞ -∞ · x1

  the pair

  0#(1(2(5(2(x1)))))

  →

  0#(x1)

  (163)

  could be deleted.

  1.1.1.1 Reduction Pair Processor

  Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

  [0#(x1)]	=	0 -∞ -∞ + -∞ 0 -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1
  [1(x1)]	=	1 0 -∞ + 0 0 0 0 0 0 -∞ 0 -∞ · x1
  [2(x1)]	=	0 -∞ -∞ + -∞ 0 0 0 0 0 0 0 0 · x1
  [1#(x1)]	=	-∞ -∞ -∞ + 0 -∞ 0 -∞ -∞ -∞ -∞ -∞ -∞ · x1
  [0(x1)]	=	-∞ 0 -∞ + 0 -∞ -∞ 0 0 0 0 -∞ -∞ · x1
  [5#(x1)]	=	0 -∞ -∞ + -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1
  [5(x1)]	=	0 -∞ 0 + -∞ -∞ -∞ -∞ -∞ 0 -∞ -∞ -∞ · x1
  [4(x1)]	=	1 0 0 + 0 0 0 0 0 0 -∞ 0 0 · x1
  [3(x1)]	=	0 -∞ -∞ + -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1

  the pair

  0#(0(1(2(x1))))

  →

  5#(0(x1))

  (106)

  could be deleted.

  1.1.1.1.1 Reduction Pair Processor

  Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

  [0#(x1)]	=	0 -∞ -∞ + -∞ 0 0 -∞ -∞ -∞ -∞ -∞ -∞ · x1
  [1(x1)]	=	0 0 1 + -∞ 0 -∞ -∞ -∞ -∞ 0 0 0 · x1
  [2(x1)]	=	0 -∞ -∞ + 0 0 0 -∞ 0 -∞ -∞ -∞ -∞ · x1
  [1#(x1)]	=	1 -∞ -∞ + -∞ 0 -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1
  [0(x1)]	=	0 -∞ -∞ + 0 0 1 -∞ 0 0 -∞ -∞ 0 · x1
  [5#(x1)]	=	1 -∞ -∞ + 0 0 0 -∞ -∞ -∞ -∞ -∞ -∞ · x1
  [5(x1)]	=	0 0 0 + -∞ -∞ 0 0 0 0 0 0 0 · x1
  [4(x1)]	=	0 -∞ -∞ + -∞ -∞ -∞ -∞ 0 -∞ -∞ -∞ -∞ · x1
  [3(x1)]	=	0 -∞ -∞ + 0 -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1

  the pairs

  5#(0(1(2(x1))))	→	5#(x1)	(148)
  5#(0(1(2(x1))))	→	1#(5(x1))	(150)

  could be deleted.

  1.1.1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 1 component.

  • The 1^st component contains the pair

    1#(0(1(2(x1))))	→	1#(x1)	(137)
    1#(0(1(2(x1))))	→	0#(x1)	(140)
    0#(1(2(x1)))	→	1#(x1)	(54)
    0#(1(2(x1)))	→	0#(x1)	(56)
    0#(1(2(x1)))	→	1#(0(x1))	(71)
    0#(1(2(x1)))	→	1#(5(x1))	(88)
    0#(0(1(2(x1))))	→	1#(x1)	(99)
    0#(0(1(2(x1))))	→	0#(x1)	(107)
    0#(1(2(0(x1))))	→	0#(0(x1))	(115)
    0#(1(4(2(2(x1)))))	→	1#(1(x1))	(171)
    0#(1(4(2(2(x1)))))	→	1#(x1)	(172)

    1.1.1.1.1.1.1 Reduction Pair Processor

    Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

    [1#(x1)]	=	1 -∞ -∞ + 0 -∞ 0 -∞ -∞ -∞ -∞ -∞ -∞ · x1
    [0(x1)]	=	0 0 -∞ + 0 0 -∞ 1 0 0 -∞ 0 -∞ · x1
    [1(x1)]	=	1 1 0 + 0 -∞ 0 0 -∞ 0 0 -∞ 0 · x1
    [2(x1)]	=	0 1 0 + 0 0 0 0 0 0 0 0 0 · x1
    [0#(x1)]	=	0 -∞ -∞ + 0 0 0 -∞ -∞ -∞ -∞ -∞ -∞ · x1
    [5(x1)]	=	1 0 1 + -∞ -∞ -∞ 0 0 -∞ -∞ -∞ -∞ · x1
    [4(x1)]	=	0 0 0 + -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ 0 · x1
    [3(x1)]	=	0 -∞ 0 + -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ 0 · x1

    the pairs

    0#(0(1(2(x1))))	→	1#(x1)	(99)
    0#(0(1(2(x1))))	→	0#(x1)	(107)

    could be deleted.

    1.1.1.1.1.1.1.1 Reduction Pair Processor

    Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

    [1#(x1)]	=	1 -∞ -∞ + 0 0 -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1
    [0(x1)]	=	-∞ -∞ 0 + 0 1 0 -∞ -∞ 0 -∞ -∞ 0 · x1
    [1(x1)]	=	0 1 1 + 0 0 0 -∞ 0 0 0 0 0 · x1
    [2(x1)]	=	0 -∞ 0 + 0 0 0 -∞ -∞ -∞ 0 0 -∞ · x1
    [0#(x1)]	=	1 -∞ -∞ + 1 0 0 -∞ -∞ -∞ -∞ -∞ -∞ · x1
    [5(x1)]	=	0 0 0 + 0 0 -∞ -∞ 0 -∞ -∞ 0 -∞ · x1
    [4(x1)]	=	0 0 0 + -∞ 0 0 -∞ -∞ -∞ -∞ 0 -∞ · x1
    [3(x1)]	=	0 0 0 + -∞ 0 0 -∞ -∞ -∞ -∞ -∞ -∞ · x1

    the pair

    1#(0(1(2(x1))))

    →

    1#(x1)

    (137)

    could be deleted.

    1.1.1.1.1.1.1.1.1 Reduction Pair Processor

    Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

    [1#(x1)]	=	0 -∞ -∞ + 0 -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1
    [0(x1)]	=	-∞ 0 0 + 0 -∞ -∞ 0 -∞ -∞ 0 -∞ -∞ · x1
    [1(x1)]	=	-∞ 0 0 + 0 -∞ 0 0 -∞ 0 0 0 0 · x1
    [2(x1)]	=	0 -∞ 1 + 0 -∞ 0 -∞ -∞ -∞ 0 1 -∞ · x1
    [0#(x1)]	=	0 -∞ -∞ + 0 0 0 -∞ -∞ -∞ -∞ -∞ -∞ · x1
    [5(x1)]	=	0 0 0 + 0 -∞ 0 0 -∞ 0 0 -∞ 0 · x1
    [4(x1)]	=	0 0 0 + -∞ -∞ -∞ -∞ -∞ 0 -∞ -∞ -∞ · x1
    [3(x1)]	=	0 -∞ -∞ + -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1

    the pair

    0#(1(2(0(x1))))

    →

    0#(0(x1))

    (115)

    could be deleted.

    1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

    Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

    [1#(x1)]	=	0 -∞ -∞ + 0 -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1
    [0(x1)]	=	0 0 0 + -∞ 0 -∞ 0 0 0 0 0 0 · x1
    [1(x1)]	=	0 0 0 + -∞ -∞ -∞ 0 0 0 0 0 0 · x1
    [2(x1)]	=	-∞ 1 0 + 0 0 0 -∞ 0 0 0 0 0 · x1
    [0#(x1)]	=	0 -∞ -∞ + 0 0 0 -∞ -∞ -∞ -∞ -∞ -∞ · x1
    [5(x1)]	=	0 0 0 + 0 0 0 0 0 0 -∞ -∞ 0 · x1
    [4(x1)]	=	0 0 0 + 0 -∞ -∞ 0 -∞ -∞ -∞ -∞ -∞ · x1
    [3(x1)]	=	0 0 0 + 0 0 0 0 0 0 0 0 0 · x1

    the pair

    0#(1(4(2(2(x1)))))

    →

    1#(1(x1))

    (171)

    could be deleted.

    1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

    Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

    [1#(x1)]	=	-∞ -∞ -∞ + 0 -∞ 0 -∞ -∞ -∞ -∞ -∞ -∞ · x1
    [0(x1)]	=	-∞ -∞ -∞ + 0 0 -∞ -∞ 1 0 0 1 -∞ · x1
    [1(x1)]	=	-∞ -∞ -∞ + 0 -∞ -∞ -∞ -∞ 0 0 -∞ -∞ · x1
    [2(x1)]	=	-∞ -∞ -∞ + 0 -∞ 0 -∞ -∞ 0 0 1 -∞ · x1
    [0#(x1)]	=	-∞ -∞ -∞ + 0 0 -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1
    [5(x1)]	=	-∞ -∞ -∞ + 0 -∞ -∞ -∞ 0 0 -∞ 0 0 · x1
    [4(x1)]	=	-∞ -∞ -∞ + 0 -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1
    [3(x1)]	=	-∞ -∞ -∞ + 0 -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1

    the pair

    1#(0(1(2(x1))))

    →

    0#(x1)

    (140)

    could be deleted.

    1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 1 component.

    • The 1^st component contains the pair

      0#(1(2(x1)))

      →

      0#(x1)

      (56)

      1.1.1.1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

      Using the linear polynomial interpretation over the naturals

      [1(x1)]	=	1 · x1
      [2(x1)]	=	1 · x1
      [0#(x1)]	=	1 · x1

      having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.

      1.1.1.1.1.1.1.1.1.1.1.1.1.1 Size-Change Termination

      Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

      0#(1(2(x1)))	→	0#(x1)	(56)
      1	>	1

      As there is no critical graph in the transitive closure, there are no infinite chains.